Spectral and scattering theory of self-adjoint Hankel operators with piecewise continuous symbols

We develop the spectral and scattering theory of self-adjoint Hankel operators $H$ with piecewise continuous symbols. In this case every jump of the symbol gives rise to a band of the absolutely continuous spectrum of $H$. We prove the existence of wave operators that relate simple `model'' (that is, explicitly diagonalizable) Hankel operators for each jump to the given Hankel operator $H$. We show that the set of all these wave operators is asymptotically complete. This determines the absolutely continuous part of $H$. We prove that the singular continuous spectrum of $H$ is empty and that its eigenvalues may accumulate only to `thresholds'' in the absolutely continuous spectrum. We also state all these results in terms of Hankel operators realized as matrix or integral operators.